GENERATING FUNCTIONS IN THE THEORY OF NUMBERS. 
145 
and the second is obtainable from the hrst by writing 
^.ry) - 
These redundant forms all lead to the same condensed form, viz. :—that derivable 
from the matrix 
(« ). 
I '-uii I 
Further we have here the most general forms of determinants such that their 
co-axial minors coincide with those of the determinant 
1 ! • 
The matrix reverts to its primary form on putting 
in the first representation, or, on putting 
^xy — l/by.^• 
in the second representation. 
The transverse matrix is obtained, from the first representation, by putting 
(ixy — l/^V, 
.ry 
Art. 50. The function V which has entered in such a fundamentally important 
manner into the foregoing analysis appears to have a place in the general theory 
of matrices. Confining ourselves, for simplicity, to the third order, it may be I’ecalled 
that Sylvester terms the function 
^ — X 
Cb3 
*^^13 
«21 
dix 
^31 
^^33 
<^'-'33 
the latent function of the matrix 
( ) 
I 
®31 *^23 
%1 ^33 <^33 
This function appears very frequently in pure mathematics, and also in applications 
to physics. From it can be derived a function of three variables, viz.- 
MDCCCXCIV.—A. U 
